Bosonic string massive scalar amplitudes and absence of "chaos''

TL;DR

The paper directly probes four-point scattering amplitudes of massive scalars in bosonic string theory, up to level 11, to assess whether higher-spin exchanges induce chaotic behavior. Using covariant formalism and a covariant $\Pi$ gauge to manage BRST-null states, the authors compute a suite of amplitudes including the first nontrivial four-identical-scalar case at level 4, and demonstrate that these amplitudes are regular and largely governed by low-spin exchanges, consistent with a simple angular-momentum model. A key methodological advance is the $\Pi$ gauge, which makes degeneracy of a given spin independent of spacetime dimension and eliminates null-state complications, aligning with Curtright’s expectations. The amplitudes are shown to factorize into the Veneziano baseline $A_{\text{Veneziano}}(s,u)$ times rational functions in $(s,u)$ with explicit polynomial numerators and denominators for multiple color orderings, revealing a dominance of spin-0 contributions and providing concrete benchmarks for higher-level and noncyclic configurations. Overall, the work furnishes explicit four-point massive-scalar benchmarks, clarifies the behavior of higher-spin exchanges in the string S-matrix, and offers a practical covariant framework for handling complex higher-level states relevant to the string-black-hole correspondence and related scattering problems.

Abstract

We compute directly in covariant formalism various four point massive scalar amplitudes in bosonic string with scalars up to level 11. The most ``difficult'' amplitude we consider is four identical scalars at level 4, i.e. the first non trivial amplitude with four identical massive scalars. All other amplitudes have at least one tachyon. All computed amplitudes are perfectly regular and show no sign of erratic behavior. While this could be naively expected it is not completely trivial because of higher spins exchanged. We give a naive argument based on a very simple model of why this can reasonably be expected and we check on the explicit examples that the intuition derived from the model is sensible. Furthermore we discuss the complications due to null states (BRST exact states) in actually computing the covariant bosonic string spectrum and how these can be overcome by using a ``$Π$ gauge'' which makes manifest that the degeneration of any spin at any mass level is independent on the spacetime dimension as predicted by Curtright et al at the end of 90s.

Figures (23)

• Figure 1: Three graphs of the functions $C_N(\theta)$ and $R_N(\theta)$ with highest frequency $N=20$
• Figure 2: Three graphs of the functions $C_N(\theta)$ and $R_N(\theta)$ with highest frequency $N=40$
• Figure 3: Some ratios $\Re \log c_N(s) / c_0(s)$ for two amplitudes "S8z1S6zS4zT" and "S8z1TS6zS4z" which can be derived from the same correlator but are quite different. In particular these amplitudes have contribution up to "spin" $16$ and $17$ and are defined for different $s$ ranges.
• Figure 4: Example of partial color ordered Veneziano amplitude $\Re \log A_{Veneziano\, 1234}(s,\theta)$ and complete $U(1)$ Veneziano amplitude $\Re \log A_{Veneziano}(s, \theta)$ when only the partial amplitude has a pole. In particular when ${\alpha'} s= 2 n$ only the color ordered partial Veneziano amplitude has a pole and this is seen by the fact that its value is bigger and resonances are smoothed out w.r.t. the complete $U(1)$ Veneziano amplitude. We have plotted the graphs with two different but small $\Im s$ in order to avoid poles and show the sensibility on the imaginary part.
• Figure 5: When when ${\alpha'} s= 2 n+1$ both amplitudes have poles and their plots are very similar.